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Power Rule
d/dx[xⁿ] = n·xⁿ⁻¹
Example: d/dx[x⁵] = 5x⁴ | d/dx[x³] = 3x² | d/dx[x] = 1 | d/dx[7] = 0
💡 Works for all n — integers, fractions, negatives. d/dx of any constant = 0 always.
If you only have 30 minutes before your calculus exam, read this in this exact order. This guide covers the 5 most tested derivative rules, 8 targeted practice problems, the 5 most common exam mistakes — and exactly how to approach problems you've never seen before.
These five rules appear on virtually every calculus exam. Do not move on until you can recall each formula from memory. If you know nothing else, know these.
Example: d/dx[x⁵] = 5x⁴ | d/dx[x³] = 3x² | d/dx[x] = 1 | d/dx[7] = 0
Example: d/dx[(3x+1)⁴] = 4(3x+1)³ · 3 = 12(3x+1)³
Example: d/dx[x²·sin x] = 2x·sin x + x²·cos x
Example: d/dx[x²/cos x] = (2x·cos x − x²·(−sin x)) / cos²x
Also via Chain Rule: d/dx[sin(u)] = cos(u)·u' | d/dx[eᵘ] = eᵘ·u'
Work through each problem in your head or on paper before expanding the solution. Spend no more than 60–90 seconds per problem. If you're stuck, expand the solution, read it carefully, then close it and redo the problem yourself.
These are the mistakes that cost students points on nearly every calculus exam. Read each one carefully — recognition alone is 90% of avoiding them during the test. For a comprehensive list, see our full derivative exam mistakes guide.
Any time you differentiate a function inside another function, you must multiply by the derivative of the inner function. This is the #1 error on calculus exams.
Students often write d/dx[cos x] = +sin x. It's always negative. This is the most-missed trig derivative on every exam.
The Quotient Rule has a minus sign in the numerator. Students frequently write u'v + uv' (copying the Product Rule). Remember: "Low d-High MINUS High d-Low."
eˣ is unique — it is its own derivative. Do not treat the exponent as a power and apply the Power Rule to it.
When differentiating a term with y with respect to x, you must attach dy/dx (Chain Rule). Forgetting this is the most common implicit differentiation error.
Hard problems are rarely hard because the math is impossible. They're hard because students panic and skip steps. These four strategies will get you through problems you've never seen before.
For deeper practice on hard problems, work through Problems 23–30 in our full practice set — focused on implicit differentiation and multi-layer chain rule. Our solution verification process is documented on our Trust & Methodology page.
Run through this list before you close this page. Click each item to check it off. If you can't honestly check all 7, go back to that section for a quick re-read.
Our free derivative calculator gives you the complete step-by-step solution for any function. Use it to check your work right now.
Open Derivative Calculator →Focus on five: the Power Rule (d/dx[xⁿ] = nxⁿ⁻¹), Chain Rule (multiply by inner derivative), Product Rule ((uv)' = u'v + uv'), Quotient Rule ((u/v)' = (u'v − uv')/v²), and the key trig derivatives: d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[tan x] = sec²x. Also memorize d/dx[eˣ] = eˣ and d/dx[ln x] = 1/x. These cover the vast majority of every calculus final exam question.
Follow this exact 30-minute plan from this guide: minutes 0–5 review the 5 most tested rules, minutes 5–15 work through 8 practice problems, minutes 15–20 study the 5 most common mistakes, minutes 20–30 review strategies for hard problems, then complete the 7-point checklist. Do not try to learn new material the night before — focus only on reinforcing what you already know. Sleep is more valuable at this point than cramming.
The single most common mistake is forgetting the Chain Rule on composite functions — writing d/dx[sin(3x)] = cos(3x) instead of 3cos(3x). The second is the wrong sign on d/dx[cos x]: it's −sin x, not +sin x. The third is using a plus sign instead of a minus in the Quotient Rule numerator. See our full derivative exam mistakes guide for every error with worked examples.
Ask these questions in order: (1) Is there a function inside another function? → Chain Rule. (2) Are two functions multiplied together? → Product Rule. (3) Is one function divided by another? → Quotient Rule. (4) Is it just xⁿ? → Power Rule. (5) Is it sin, cos, tan, eˣ, or ln x? → use their specific formulas. Many hard problems combine two rules — Product Rule on the outside, Chain Rule on each part inside. Use our basic derivative calculator to verify your rule identification.
Most university calculus exams do not permit derivative calculators, or restrict to basic arithmetic only. Our free derivative calculator is designed for practice verification — use it after attempting problems yourself to check work and understand mistakes, not as a substitute for learning the rules. The problems on your exam are designed to test your hand-calculation skills.
Every problem and solution on this page has been reviewed by our mathematics education team — experienced calculus instructors and university-level educators who understand what students face on finals. Our review and trust methodology ensures all mathematical content is accurate, pedagogically sound, and aligned with standard university calculus curricula. Found an error? Contact us — we review every report within 24 hours.